What is a Half-Life?
A half-life (t_{1/2}) is the amount of time it takes for exactly 50% (12) of a radioactive isotope's unstable nuclei to decay into a stable daughter element.
No matter how large or small your sample is, after one half-life passes, half of it is gone:
- •0 Half-lives: 100% remains (11)
- •1 Half-life: 50% remains (12)
- •2 Half-lives: 25% remains (14)
- •3 Half-lives: 12.5% remains (18)
- •4 Half-lives: 6.25% remains (116)
The Two Core Formulas
1. Number of Half-Lives (n)
2. Remaining Amount (N_t)
- •N_0 = Initial mass or count
- •N_t = Remaining mass or count
- •n = Number of half-lives that elapsed
Worked Example 1: Finding Remaining Mass with the Table Method
The Table Method is the fastest and most foolproof strategy for standard classroom exams.
Problem:
Iodine-131 has a half-life of 8 days. A hospital receives a shipment of 64 mg.
How much Iodine-131 remains after 32 days?
Step 1: Calculate the number of half-lives (n)
Step 2: Build the Decay Table
Set up a table with half-life intervals and cut the mass in half at each step:
| Half-Lives (n) | Time Elapsed (Days) | Remaining Mass (mg) | Fraction Left |
|---|---|---|---|
| 0 (Start) | 0 days | 64 mg | 1 (100%) |
| 1 | 8 days | 64 ÷ 2 = \mathbf{32 mg} | 12 (50%) |
| 2 | 16 days | 32 ÷ 2 = \mathbf{16 mg} | 14 (25%) |
| 3 | 24 days | 16 ÷ 2 = \mathbf{8 mg} | 18 (12.5%) |
| 4 | 32 days | 8 ÷ 2 = \mathbf{4 mg} | 116 (6.25%) |
After 32 days, 4 mg of Iodine-131 remains.
Worked Example 2: Working Backwards to Find Initial Mass (N_0)
What if the exam asks for how much sample you started with? Work the process in reverse—double the mass for each half-life!
Problem:
After 48 hours, only 2.5 grams of Technetium-99m remains.
The half-life is 6 hours. What was the original starting mass?
Step 1: Calculate elapsed half-lives
Step 2: Work backwards (Multiply by 2 for each step)
Instead of dividing by 2, we multiply by 2 eight times (28):
The hospital originally started with 640 grams.
Fraction and Percentage Reference Table
| Elapsed Half-Lives (n) | Fraction Remaining | Percentage Remaining | Percentage Decayed Away |
|---|---|---|---|
| 1 | 12 | 50% | 50% |
| 2 | 14 | 25% | 75% |
| 3 | 18 | 12.5% | 87.5% |
| 4 | 116 | 6.25% | 93.75% |
| 5 | 132 | 3.125% | 96.875% |
Careful with Question Wording:
Always read whether the question asks:
1. "How much remains?" (Use N_t directly: e.g., 4 mg)
2. "How much decayed away?" (Subtract from starting amount: 64 - 4 = 60 mg decayed!)